Linear and Nonlinear Elliptic and Parabolic Partial Differential Equations

Lecturer

Dr Yann Bernard, Monash University

Synopsis

This graduate-level course provides a rigorous and comprehensive introduction to second-order partial differential equations, with a primary focus on linear elliptic and parabolic equations. The course develops both the classical theory (maximum principles, regularity, fundamental solutions) and the modern weak formulation using Sobolev spaces and variational methods. The final week introduces classical nonlinear models, including semilinear elliptic and reaction–diffusion equations, emphasizing tools like sub- and supersolution methods and maximum principles. The course is designed to balance theory with concrete examples and includes weekly exercise sessions to reinforce core techniques.

Course Overview

Week 1: Foundations and Classical Theory of Elliptic Equations

• Introduction to PDEs and Classification
• Classical Theory of Linear Elliptic Equations
• Properties of Harmonic Functions and Classical Solutions

Week 2: Weak Solutions and Sobolev Spaces 

• Introduction to Sobolev Spaces
• Weak Formulation of Elliptic Boundary Value Problems
• Elliptic Regularity in the Sobolev Framework
  

Week 3: Linear Parabolic Equations and Schauder Theory

• Linear Parabolic Equations and Weak Formulation
• Parabolic Regularity and Smoothing Effects
• Schauder Estimates for Elliptic and Parabolic PDEs

Week 4: Classical Nonlinear Elliptic and Parabolic Equations

• Classical Nonlinear Elliptic Equations
• Reaction–Diffusion Equations and Local Existence
• Nonlinear Diffusion and Asymptotic Behaviour

Prerequisites